Recursion-Theoretic Ranking and Compression

TL;DR

This paper explores recursion-theoretic versions of ranking and compression functions, analyzing which sets admit such functions and their properties, inspired by complexity theory but within computability theory.

Contribution

It introduces and studies recursion-theoretic notions of ranking and compression functions, connecting them to 1-truth-table degrees and coRE cylinders.

Findings

Rankable and compressible sets broadly populate the 1-truth-table degrees.

Every nonempty coRE cylinder is recursively compressible.

Abstract

For which sets A does there exist a mapping, computed by a total or partial recursive function, such that the mapping, when its domain is restricted to A, is a 1-to-1, onto mapping to $\Sigma^*$? And for which sets A does there exist such a mapping that respects the lexicographical ordering within A? Both cases are types of perfect, minimal hash functions. The complexity-theoretic versions of these notions are known as compression functions and ranking functions. The present paper defines and studies the recursion-theoretic versions of compression and ranking functions, and in particular studies the question of which sets have, or lack, such functions. Thus, this is a case where, in contrast to the usual direction of notion transferal, notions from complexity theory are inspiring notions, and an investigation, in computability theory. We show that the rankable and compressible sets broadly populate the 1-truth-table degrees, and we prove that every nonempty coRE cylinder is recursively compressible.